U.S. patent application number 14/034521 was filed with the patent office on 2015-03-26 for computationally efficient nonlinear structural analysis.
This patent application is currently assigned to SAN DIEGO STATE UNIVERSITY RESEARCH FOUNDATION. The applicant listed for this patent is SAN DIEGO STATE UNIVERSITY RESEARCH FOUNDATION. Invention is credited to ROBERT DOWELL.
Application Number | 20150088428 14/034521 |
Document ID | / |
Family ID | 52691684 |
Filed Date | 2015-03-26 |
United States Patent
Application |
20150088428 |
Kind Code |
A1 |
DOWELL; ROBERT |
March 26, 2015 |
COMPUTATIONALLY EFFICIENT NONLINEAR STRUCTURAL ANALYSIS
Abstract
Seismic displacement demands for design of a bridge frame
structure are typically determined from linear-elastic analysis
(LEA), which are often incorrect, and compared to displacement
capacity from a nonlinear pushover analysis. Nonlinear time-history
analysis (NTHA) provides the most realistic assessment of
displacement demands because it properly models the physics of the
dynamic problem, wherein stiffness of the bridge varies over time.
However, using NTHA to determine a bridge response from multiple
earthquake motions based on the stiffness method requires excessive
time. A unique approach for determining the nonlinear time-history
response of a bridge frame is disclosed that is thousands of times
faster than the stiffness method while providing the same results.
Computational efficiency allows bridge design engineers to use NTHA
for the seismic design of bridge structures by producing multiple
determinations in less than one second. Displacement demands and
capacities are based on nonlinear bridge behavior, resulting in
safer bridge structures and reduced construction costs.
Inventors: |
DOWELL; ROBERT; (San Diego,
CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SAN DIEGO STATE UNIVERSITY RESEARCH FOUNDATION |
San Diego |
CA |
US |
|
|
Assignee: |
SAN DIEGO STATE UNIVERSITY RESEARCH
FOUNDATION
San Diego
CA
|
Family ID: |
52691684 |
Appl. No.: |
14/034521 |
Filed: |
September 23, 2013 |
Current U.S.
Class: |
702/14 |
Current CPC
Class: |
E01D 1/00 20130101; E01D
19/00 20130101; G06F 30/13 20200101; G01M 5/0041 20130101; G06F
2111/10 20200101; G01M 5/0008 20130101 |
Class at
Publication: |
702/14 |
International
Class: |
G01V 1/30 20060101
G01V001/30 |
Claims
1. A method for seismic analysis of frame structures comprising:
performing an initial dead load analysis of structure moments and
stiffness; calculating, from Incremental Closed Form Method (ICFM)
Equations, incremental time values of acceleration, velocity,
displacement and final structure moments; summing the incremental
values to produce a total sum value of all calculated time
increment values; adjusting frame stiffness values are for a next
incremental time value calculation; scaling the calculated time
increment values for the time increment to the time of an event;
and repeating the calculating, summing, adjusting and scaling until
ICFM calculations of all time increments of ground motion have been
completed and summed.
2. The method of claim 1 wherein, R=Cycle factor going to the right
of a beam, T=Cycle factor going to the left of a beam,
r=Distribution factor for member on the right side of a joint,
t=Distribution factor for member on the left side of a joint,
c=Distribution factor for column at a joint, AB.sub.C=Member moment
just to the right of Joint A from a unit moment applied at Joint B
for a continuous beam or bridge frame with C number of internal
joints, BA.sub.C=Member moment just to the left of a Joint B from a
unit moment applied at a Joint A for a continuous beam or bridge
frame with C number of internal joints,
r.sub.Ar.sub.B=Multiplication of r.sub.A, r.sub.A+1, . . . through
r.sub.B, r.sub.2r.sub.5=Multiplication of r.sub.2, r.sub.3, r.sub.4
and r.sub.5, R.sub.AR.sub.B=Multiplication of R.sub.A, R.sub.A+1, .
. . through R.sub.B, t.sub.At.sub.B=Multiplication of t.sub.A,
t.sub.A+1, . . . through t.sub.B, and T.sub.AT.sub.B=Multiplication
of T.sub.A, T.sub.A+1, . . . through T.sub.B.
3. The method of claim 2 wherein, A Superstructure Right Moment is
defined as AB C = r B . r A ( - 2 ) A - B R B R C T C T A + 1 [ 1 -
t A + 1 4 T A + 1 ] . ##EQU00009##
4. The method of claim 2 wherein, a Column Right Moment is defined
as AB C = r B r A - 1 c A ( - 2 ) A - B R B R C T C T A + 1
##EQU00010##
5. The method of claim 2 wherein, a Simplified Superstructure Right
Moment for a Last Internal Joint is defined as CB C = r B r C ( - 2
) C - B R B R C . ##EQU00011##
6. The method of claim 2 wherein, a Simplified Column Right moment
for a Last Internal Joint is defined as CB C = r B r C - 1 c c ( -
2 ) C - B R B R C . ##EQU00012##
7. The method of claim 2 wherein, a Superstructure Left Moment is
defined as BA C = t A t B ( - 2 ) A - B T 1 T A R 1 R B - 1 [ 1 - r
B - 1 4 R B - 1 ] ##EQU00013##
8. The method of claim 2 wherein, a Column left Moment is defined
as BA C = t A t B + 1 c B ( - 2 ) A - B T 1 T A R 1 R B - 1
##EQU00014##
9. The method of claim 2 wherein, a Simplified Superstructure Left
Moment for a First Internal Joint is defined as 1 A C = t 1 t A ( -
2 ) A - 1 T 1 T A ##EQU00015##
10. The method of claim 2 wherein, A Simplified Column Left Moment
For a First Internal Joint is defined as 1 A C = t 2 t A c 1 ( - 2
) A - 1 T 1 T A ##EQU00016##
11. The method of claim 1 wherein the method is implemented in a
mobile application or mobile device.
12. A computer readable medium having instructions stored thereon
to cause a processor in a wireless device to: perform an initial
dead load analysis of structure moments and stiffness; calculate,
from Incremental Closed Form Method (ICFM) Equations, incremental
time values of acceleration, velocity, displacement and final
structure moments; sum the incremental values to produce a total
sum value of all calculated time increment values; adjust frame
stiffness values are for a next incremental time value calculation;
scale the calculated time increment values for the time increment
to the time of an event; and repeat the calculating, summing,
adjusting and scaling until ICFM calculations of all time
increments of ground motion have been completed and summed.
13. The computer readable medium of claim 12 wherein, R=Cycle
factor going to the right of a beam, T=Cycle factor going to the
left of a beam, r=Distribution factor for member on the right side
of a joint, t=Distribution factor for member on the left side of a
joint, c=Distribution factor for column at a joint, AB.sub.C=Member
moment just to the right of Joint A from a unit moment applied at
Joint B for a continuous beam or bridge frame with C number of
internal joints, BA.sub.C=Member moment just to the left of a Joint
B from a unit moment applied at a Joint A for a continuous beam or
bridge frame with C number of internal joints,
r.sub.Ar.sub.B=Multiplication of r.sub.A, r.sub.A+1, . . . through
r.sub.B, r.sub.2r.sub.5=Multiplication of r.sub.2, r.sub.3, r.sub.4
and r.sub.5, R.sub.AR.sub.B=Multiplication of R.sub.A, R.sub.A+1, .
. . through R.sub.B, t.sub.At.sub.B=Multiplication of t.sub.A,
t.sub.A+1, . . . through t.sub.B, and T.sub.AT.sub.B=Multiplication
of T.sub.A, T.sub.A+1, . . . through T.sub.B.
14. The computer readable medium of claim 13 wherein, A
Superstructure Right Moment is defined as AB C = r B r A ( - 2 ) A
- B R B R C T C T A + 1 [ 1 - t A + 1 4 T A + 1 ] .
##EQU00017##
15. The computer readable medium of claim 13 wherein, a Column
Right Moment is defined as AB C = r B r A - 1 c A ( - 2 ) A - B R B
R C T C T A + 1 . ##EQU00018##
16. The computer readable medium of claim 13 wherein, a Simplified
Superstructure Right Moment for a Last Internal Joint is defined as
CB C = r B r C ( - 2 ) C - B R B R C . ##EQU00019##
17. The computer readable medium of claim 13 wherein, a Simplified
Column Right moment for a Last Internal Joint is defined as CB C =
r B r C - 1 c c ( - 2 ) C - B R B R C . ##EQU00020##
18. The computer readable medium of claim 13 wherein, a
Superstructure Left Moment is defined as BA C = t A t B ( - 2 ) A -
B T 1 T A R 1 R B - 1 [ 1 - r B - 1 4 R B - 1 ] ##EQU00021##
19. The computer readable medium of claim 13 wherein, a Column left
Moment is defined as BA C = t A t B + 1 c B ( - 2 ) A - B T 1 T A R
1 R B - 1 ##EQU00022##
20. The computer readable medium of claim 13 wherein, a Simplified
Superstructure Left Moment for a First Internal Joint is defined as
1 A C = t 1 t A ( - 2 ) A - 1 T 1 T A ##EQU00023##
21. The computer readable medium of claim 13 wherein, A Simplified
Column Left Moment For a First Internal Joint is defined as 1 A C =
t 2 t A c 1 ( - 2 ) A - 1 T 1 T A ##EQU00024##
Description
CLAIM OF PRIORITY UNDER 35 U.S.C. .sctn.119
[0001] The present application for Patent claims priority to
Provisional Application No. 61/705,140 entitled "Nonlinear
Incremental Closed-Form Method for Frame Structures" filed Sep. 24,
2012, and assigned to the assignee hereof and hereby expressly
incorporated by reference herein.
BACKGROUND
[0002] 1. Field
[0003] The present invention relates generally to seismic bridge
design, and more specifically to computationally efficient
nonlinear time-history analysis for determination of bridge frame
response and demands.
[0004] 2. Background
[0005] Nonlinear seismic time-history analysis for bridge frame
structures is traditionally performed using an incremental
stiffness method. For each small time increment (typically between
0.005 and 0.02 seconds), a change in ground acceleration is applied
to the base of the bridge structure. Incremental member forces as
well as displacements are determined by solving a system of
simultaneous equations using matrix mathematics. The total response
of the structure at any time during the earthquake is found by
summing all prior incremental results.
[0006] This approach can require one or more hours for one
earthquake analysis. A full measured earthquake record may contain
accelerations at 10,000 time increments, and because of
nonlinearities in the response, the complete bridge structure must
be solved at each of these increments. Therefore, all of the
simultaneous equations required for a single loading case are
solved 10,000 times in a row, resulting in excruciatingly slow
computation times. Additionally, due to the numerical nature of the
stiffness method, iteration is often required to satisfy
equilibrium when nonlinearities occur.
[0007] Computer nonlinear structural analysis algorithms based on
the stiffness method frequently malfunction while performing
iterations associated with severe nonlinearities and, because
subsequent results are a summation of all prior results, produce no
results for the remainder of the earthquake record. Because these
severe nonlinearities occur at maximum earthquake shaking, the
traditional matrix approach usually provides accurate response
results only from small earthquake shaking at the start of the
earthquake. Due to the extremely slow computational speed and
iteration failures, current nonlinear time-history analysis methods
are not practical for commonplace bridge design. Especially since
each earthquake analysis can take up to an hour or more, and
multiple earthquakes analyses are required to allow for variability
of future seismic events.
[0008] One skilled in the art would recognize that building
structures are analyzed in much the same way as bridges. Ground
motions are provided as accelerations to the base of the building
at multiple time increments and the nonlinear response of the
building is found using time-history analysis with the stiffness
method to determine incremental results. Total results at any point
during the earthquake are found from summing all prior incremental
results. As with bridge frame structures discussed above, multiple
simultaneous equations must be solved at each time increment. The
same difficulties exist for analyzing building frame and other
structures under severe seismic motion. There is, therefore, a need
in the art for a highly time-efficient and cost-effective method
for determining an exact solution for structural motion response
using simple equations, which can provide a worldwide commonplace
tool for bridge and other structural design.
SUMMARY
[0009] Embodiments disclosed herein address the above stated needs
by providing a method for generating an exact structural analysis
under any conditions by disclosing a convergent geometric series
and taking it to the limit with calculus to produce a high speed
exact solution with simple closed-form equations. This method
provides an exact solution without failing under nonlinear
conditions by incrementally summing non-numerical results from
individual closed form equations to provide a total result at all
times.
[0010] A novel incremental closed-form method (ICFM) solves the
incremental member forces through analytical equations and in this
sense, at each time increment, this is not a numerical approach. It
is this analytical solution that gives (1) the method its stability
and (2) its computational speed, since no simultaneous equations
are required as in the stiffness method. Total results are found by
summing all of the incremental results up to a given time. While
other known methods depend on matrix mathematics, the closed-form
equations used in the ICFM require no matrices or vectors and yet
provide exact results. The new ICFM can be viewed as a combination
of analytical and numerical approaches to determine the nonlinear
time-history response of a frame structure subjected to earthquake
motions. Numerical analysis is used to advance the displacement,
velocity and acceleration results of the structure for each time
increment while the analytical (closed-form) equations are used to
find the exact change in member forces that occurred over the same
time increment.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] FIG. 1 illustrates exemplary bridge frame structures for
computationally efficient nonlinear structural analysis;
[0012] FIG. 2 illustrates exemplary incremental column fixed
end-moments of a bridge structure for computationally efficient
nonlinear structural analysis;
[0013] FIG. 3 illustrates exemplary final incremental column and
superstructure end-moments of a bridge structure for
computationally efficient nonlinear structural analysis;
[0014] FIG. 4 illustrates exemplary column nonlinear
moment-rotation springs for computationally efficient nonlinear
structural analysis;
[0015] FIG. 5 illustrates exemplary Time Shift and Increment
Renumbering for computationally efficient nonlinear structural
analysis;
[0016] FIG. 6A is an exemplary high level flow chart illustrating
the computationally efficient Incremental Closed Form Method (ICFM)
for nonlinear structural analysis;
[0017] FIG. 6B is an exemplary detailed flowchart illustrating the
computationally efficient Incremental Closed Form Method (ICFM) for
nonlinear structural analysis;
[0018] FIG. 7 illustrates a bridge frame superstructure in an
exemplary nonlinear time-history analysis using a computationally
efficient nonlinear structural analysis algorithm;
[0019] FIG. 8 details an exemplary bridge frame superstructure
cross-section geometry in an exemplary nonlinear time-history
analysis using a computationally efficient nonlinear structural
analysis algorithm;
[0020] FIG. 9 details exemplary column plastic moments of a bridge
frame superstructure in an exemplary nonlinear time-history
analysis using a computationally efficient nonlinear structural
analysis algorithm;
[0021] FIG. 10 details the exemplary bridge frame measured ground
acceleration time-history at one station of the 1989 Loma Prieta
earthquake;
[0022] FIG. 11 details the exemplary bridge frame ground
displacement time-history at one station of the 1989 Loma Prieta
earthquake;
[0023] FIG. 12 details the exemplary bridge frame acceleration
response spectrum at one station of the 1989 Loma Prieta
earthquake;
[0024] FIG. 13 details the exemplary bridge frame computationally
efficient structural analysis algorithm linear-elastic
force-displacement results from the 1989 Loma Prieta
earthquake;
[0025] FIG. 14 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear
force-displacement results from the 1989 Loma Prieta
earthquake;
[0026] FIG. 15 details the exemplary bridge frame computationally
efficient structural analysis algorithm linear-elastic displacement
time-history response results from the 1989 Loma Prieta
earthquake;
[0027] FIG. 16 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear displacement
time-history response results from the 1989 Loma Prieta
earthquake;
[0028] FIG. 17 details the exemplary bridge frame computationally
efficient structural analysis algorithm linear-elastic force
time-history results from the 1989 Loma Prieta earthquake;
[0029] FIG. 18 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear force
time-history response results from the 1989 Loma Prieta
earthquake;
[0030] FIG. 19 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear displacement
response results (between 5 and 15 s zoom-in) from the 1989 Loma
Prieta earthquake;
[0031] FIG. 20 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear force response
results (between 5 and 15 s zoom-in) from the 1989 Loma Prieta
earthquake.
DETAILED DESCRIPTION
[0032] The word "exemplary" is used herein to mean "serving as an
example, instance, or illustration." Any embodiment described
herein as "exemplary" is not necessarily to be construed as
preferred or advantageous over other embodiments.
[0033] The disclosed incremental closed-form approach to structural
analysis is fundamentally different from previously known stiffness
and moment distribution methods, wherein no simultaneous equations
are required as in the stiffness method and moments need not be
distributed back-and-forth as in moment distribution. At each time
increment an exact incremental bridge or other structural response
is found from simple closed-form equations. This novel method is
exponentially faster than the incremental stiffness method,
allowing a complete nonlinear earthquake analysis to be generated
in a fraction of a second rather than up to an hour or more. Two
example bridge analyses with different ground motions have shown
the new method to be over 3,000 and over 5,000 times faster than
the stiffness method. Advantageously, the present approach is
stable under nonlinear conditions because results are determined
from closed-from (analytical) equations without requiring matrix
manipulations and without iterations. At each time increment, an
algorithm monitors, for example, all column ends of a bridge for
nonlinear behavior, as bridge frames are designed to allow
nonlinear member behavior only at the column ends. If a nonlinear
event occurs (i.e. plastic hinge) within the time increment, the
algorithm determines the time it occurred and backs all results to
this point, and changes the stiffness of the moment-rotation hinge
at the column end, which results in a change to the frame
stiffness. A new point is provided at each nonlinear event. There
is no iteration needed to provide the correct results at the
nonlinear event. Furthermore, there is no force overshoot before or
after the event.
[0034] The algorithm also monitors the structure for unloading
behavior. For example, when a bridge frame changes direction, the
stiffness values of the plastic hinges and structure also change,
affecting the response. If a direction change occurs within a time
increment, the algorithm determines when this happened and returns
all results to this time, providing an additional data point. The
stiffness is changes and the solution scheme continues. Thus, no
iteration is required.
[0035] Because the incremental closed-form approach is highly
time-efficient and completely stable, ensuring that results are
found for the full duration of the seismic event, multiple
earthquake motions (for example 10 different average earthquake
records) can be analyzed for a given bridge or other structure in
approximately one to three seconds (fraction of a second per
earthquake analysis). This novel method provides for nonlinear
time-history analysis for cost-effective and time-efficient
structural design, replacing current linear-elastic analysis
methods, which cannot capture physical behavior under severe
earthquake loading. It is important to note that different design
codes (specifications) that govern the analysis and design of
bridges and buildings do allow for more sophisticated nonlinear
time-history analysis, but because they are highly inefficient and
time-consuming, nonlinear time-history analysis is unfeasible for
commonplace applications, which is very unfortunate as bridges and
buildings are designed to respond to moderate and large earthquakes
in the nonlinear range. Providing this new analysis tool allows
structural design engineers to correctly assess the seismic
behavior of various structures, making them safer and often less
costly to construct.
[0036] The present invention resolves the three primary reasons
that nonlinear time-history analysis (NTHA) is not performed for
everyday bridge design in high seismic regions, namely, (1) that
the analysis takes too long, (2) it often stops running due to
numerical difficulties at the time of largest shaking, providing no
results at or beyond this point of most interest and (3) the future
earthquake motion that the bridge will be subjected to is unknown.
Traditionally, displacement demands are found from linear-elastic
spectral analysis (LESA) and compared to displacement capacity
determined from a nonlinear pushover analysis. This approach is
loosely based on the equal displacement principle, which states
that linear-elastic and nonlinear seismic displacement demands are
approximately equal, so long as the initial stiffness for nonlinear
analysis is the same as the linear-elastic stiffness. While maximum
displacements may be similar between LESA and NTHA, force demands
from linear-elastic analysis are typically many times larger than
from nonlinear analysis due to column-end plastic hinges that limit
the force levels. It is widely accepted that NTHA is far superior
to LESA and linear-elastic time-history analysis (LETHA), as it
recognizes the proper physics of the problem, with changing
stiffness as plastic hinges form and cycle at the ends of the
columns, which they are designed and detailed to do.
[0037] The present algorithm comprises closed-form equations used
to determine incremental forces of a redundant bridge frame that
develop from each time step of the earthquake record. In this
sense, the incremental closed-form method (ICFM) is not
numerically-based but analytical, at least at each time step,
resulting in exact force values and a stable analysis. Total
results at all times are found by summing the prior incremental
results. No simultaneous equations are required, resulting in the
extremely fast analysis presented here. Nonlinear behavior can only
occur at the columns ends, which is consistent with bridge design
practice. All other structural members are protected from going
nonlinear by capacity-design principles. Any number of spans can be
analyzed (from one to infinity).
[0038] As mentioned above, one often-cited reason for not using
NTHA for seismic bridge design is that the future earthquake motion
is not yet known. Therefore, many different earthquake motion
analyses would need to be conducted in order to reveal all possible
structural responses. Because a single NTHA calculation requires
excessive time using the traditional stiffness method,
consideration of multiple analyses becomes daunting for commonplace
design purposes. Due to the sheer speed and computational
efficiency of the novel approach presented here (ICFM), many
different earthquake motions can be considered and maximum (or
average) responses automatically collected and presented.
Calculating 10, 100 or even 1,000 NTHAs to support a new bridge
design using the proposed ICFM presents no difficulty, essentially
placing the power of a supercomputer on each user's desk.
[0039] FIG. 1 illustrates exemplary bridge frame structures 100 for
computationally efficient nonlinear structural analysis wherein
bridge 102 comprises five spans, bridge 104 comprises four spans
and bridge 106 comprises 3 spans. Computationally efficient
nonlinear structural analysis is advantageously applied to any
number of bridge spans. A bridge structure is used herein and
throughout while one skilled in the art would understand that
computationally efficient nonlinear structural analysis can be
applied to any type of frame structure. The extremely fast and
stable seismic analysis of a bridge frame is realized in
computationally efficient nonlinear structural analysis by
combining (1) incremental closed-form equations, (2) event-scaling
analysis, (3) capacity design principles and (4) the average
acceleration method. The computationally efficient nonlinear
structural analysis algorithm incorporates approaches (1)-(4) to
determine the response over time of a bridge frame that can develop
plastic hinges at its column ends.
[0040] FIG. 2 illustrates exemplary bridge frame structure 200,
column fixed-end-moments 202a-d for computationally efficient
nonlinear structural analysis. The computationally efficient
nonlinear structural analysis ICFM algorithm gives a displacement
increment 204a-c to the bridge frame structure 200 while the joints
are fixed from rotation, resulting in incremental fixed-end-moments
at the column ends 202a-d and incremental out-of-balance moments
202a-b at the internal joints 202e-f. The algorithm's closed-form
equations are then used to determine final incremental
member-end-moments 202a-d (in FIG. 3) and 302a-d (in FIG. 3)
associated with this displacement increment 204a-c of the frame
structure 200. Note that 202a-d in FIG. 3 have different values
than 202a-d in FIG. 2. In FIG. 2 they are the column
fixed-end-moments (internal joints fixed from rotation) and in FIG.
3 they are the final moments after the internal joints have been
allowed to rotate to find equilibrium. Each change in displacement
204a-c is found from an equation of motion, using the stable
average acceleration method. Each displacement increment is based
on the current stiffness of the frame 200, including flexibility of
the superstructure and columns. The closed-form equations were
derived directly from the logic of moment distribution and
calculus.
[0041] FIG. 3 illustrates exemplary final incremental column and
superstructure end-moments of a bridge structure for
computationally efficient nonlinear structural analysis. Moments
are found from the incremental closed-form equations while holding
the frame 200 at the same incremental sway displacement level
204a-c that developed the fixed-end-moments shown in FIG. 2,
resulting in final incremental moments at the superstructure and
column ends 202a-d and 302a-d. Once the member-end-moments are
known, all other member forces (shear forces/axial forces) and
reactions are readily found from statics.
[0042] FIG. 4 illustrates exemplary column nonlinear
moment-rotation springs 400. Columns 206a-b are modeled as beam
members with nonlinear moment-rotation springs at both ends 402a-b
such that elasto-plastic behavior, or the more complex behavior
defined by the Takeda or Pivot Hystgersis models, can readily be
analyzed by the computationally efficient nonlinear structural
analysis algorithm.
[0043] FIG. 5 illustrates exemplary Time Shift and Increment
Renumbering for computationally efficient nonlinear structural
analysis 500. The speed and stability of the computer
computationally efficient nonlinear structural analysis algorithm
result from the use of (1) the closed-form equations at each time
increment to determine the change in member-end-moments, with no
simultaneous equations or numerical analyses required, and (2)
event-scaling when a column moment has exceeded the plastic moment
capacity of one of the column ends. All moments and time values are
scaled back to the exact values when the event developed, with no
iteration necessary and no force overshoot. For each time increment
the response of the structure is linear, and to allow for nonlinear
behavior the results are scaled back to the exact point when a
nonlinear event has occurred. The stiffness of the system is
changed (recognizing nonlinear behavior) and then the structure
response is found for the next time increment, also as a linear
response. In FIG. 5, the increment times are initially 0.02 s, but
between increment numbers 262 and 263 a nonlinear event develops.
Using the equations from the average acceleration method, a simple
and exact solution for the time when this event occurred was
derived. In this example, the nonlinear event occurs 0.005 s into
the full time step of 0.02 s. A new increment step is provided at
the time of the nonlinear event, and all future increments are
relabeled (this is indicated by the diagonal line across increment
numbers 263 and 264 and new labels 263 through 265). Forces are
linearly scaled back to the time of the event, resulting in one
column moment that exactly equals its plastic moment capacity and
all other column moments that are below their capacities. An
additional data point 506 is provided at the time the event
occurred, requiring a reduced time step to that point and another
reduced time step to the end of the original full time step. In the
example provided in FIG. 5, the nonlinear event happened between
steps 262 and 263 (502) at 5.245 s, with the full time step of 0.02
s reduced to 0.005 s. A new point is provided at the time of the
event, with this now defined as step 263 (506), requiring all
further steps to be renumbered. After the reduced time step of
0.005 s to reach the event, a second reduced time step of 0.015 s
is used to make up a full time step of 0.02 s and get to the new
step 264 (502) at time 5.26 s. Until another event occurs, the
analysis continues from step-to-step with full time steps of 0.02
s. If additional events develop within one full time step, the same
process is followed.
[0044] Often more than one plastic hinge occurs at the same time
(within a given tolerance). In this case, the computationally
efficient nonlinear structural analysis algorithm allows multiple
nonlinear hinges to develop as one event. Ground accelerations must
also be scaled (interpolated) to agree with the modified time
steps. The stiffness is changed following scale-back to reflect the
new frame condition of a pin holding a plastic moment with no
rotational stiffness (no additional moment capacity while the hinge
is free to rotate) at the plastic hinge location. When a
displacement reversal occurs, the time of reversal is found (also
based on an exact equation derived from the average acceleration
method) and the event-scaling procedure is employed so that the
correct stiffness is included for the reduced time step following
reversal. If a reversal happens when there is no nonlinear behavior
then there is no scale-back required, as the stiffness will not
change.
[0045] Bridges are designed to force nonlinear behavior into the
column ends while protecting the superstructure and footings from
inelastic behavior. This is achieved by providing more strength to
these members than can be induced by column plastic hinges based on
capacity design principles. Column end regions are specially
designed and detailed to allow large plastic rotations, which
translate to large displacement capacity of the bridge frame. Prior
to seismic analysis, the closed-form equations are used to
determine superstructure and column-end-moments from the
self-weight of the bridge. Potential gravity sidesway is included
in determination of dead load moments. The structural analysis
algorithm calculates no-sway dead load moments from the closed-form
equations. A second application of the closed-form equations from a
scaled lateral sway provides the dead load sway moments, with total
dead load moments equal to the sum of the no-sway and sway values.
Dead load results are verified by checking that the column base
shears for the entire frame sum to zero. This is the condition the
bridge frame will be in at the time of an earthquake, and so the
existing dead load column moments must be included in the seismic
analysis as they affect how much additional moment is required to
cause plastic hinging. The new computationally efficient nonlinear
structural analysis algorithm based on the ICFM automatically
considers this.
[0046] The explicit form of the average acceleration method (a
Newmark-Beta method) was used to advance the solution to the next
time increment of the earthquake. In one exemplary embodiment, this
specific Newmark-Beta method is used because it is unconditionally
stable and because it requires no iteration to move from
step-to-step. At the start of an earthquake, the initial stiffness
of the bridge frame is used in the equations of motion. This
stiffness will change only following an event, including the
formation or unloading of a plastic hinge. When an event has
occurred, all of the results are scaled back to the time when the
event first happened, without iteration. The stiffness of the frame
is changed and the equations from the average acceleration method
are used to move to the next time increment.
[0047] Following a nonlinear event, and before scaling results back
to the time of this event, the moment-rotation stiffness of the
plastic hinge is changed in the computationally efficient nonlinear
structural analysis algorithm and one additional full time step is
taken to determine the reduced lateral stiffness of the bridge
frame. The sum of the base shears represents the change in
restoring force of the frame and the new stiffness is this force
divided by the incremental displacement. Until another nonlinear
event occurs, this is the stiffness that is used in the various
equations of the explicit form of the average acceleration
method.
[0048] At the end of each time increment, the average acceleration
method provides incremental displacement, velocity and acceleration
values of the bridge frame, which are added to the prior results to
obtain total values up to that point in time, ultimately giving
complete time-history responses for the earthquake. Incremental
column moments are added to the prior values to obtain total column
moments at the end of the time increment. This process continues
until a column moment exceeds the plastic moment capacity of one of
the column ends, where the moments are largest. Column and
superstructure moments, as well as the frame displacement, are
scaled back to their values at the time of the nonlinear event.
[0049] The stiffness of the frame changes following each nonlinear
event, representing the plastic hinge that has occurred, and a new
incremental response is found, continuing until all time increments
for the base motion have completed. If the relative frame
displacement reverses, the time when this reversal happens is
calculated, without iteration, and all relevant results are scaled
back to the appropriate values. Upon frame displacement reversal,
all plastic hinges start unloading, returning to their rigid state,
and the original frame stiffness is used until another plastic
hinge has formed. Multi-degree-of-freedom nonlinear seismic
analysis is conducted (multiple displacement DOF, including joint
rotations and frame translation) using the ICFM for a bridge frame
with a single mass degree-of-freedom representing the mass of the
entire frame. Once the complete closed-form method is finalized,
which will include member axial deformations, then multiple mass
degrees-of-freedom are also possible.
[0050] FIG. 6A is an exemplary high level flow chart illustrating
the computationally efficient Incremental Closed Form Method (ICFM)
for nonlinear structural analysis 600A where frame structure forces
are analytically (non-numerically) determined using present novel
ICFM equations for each time increment of the analysis and then
numerically summing the incremental moment values to produce a
total sum value structural analysis.
[0051] Processing begins in step 602, where an initial dead load
analysis of structure moments is performed and initial lateral
stiffness determined Control flow proceeds to step 604.
[0052] In step 604, an increment of acceleration, velocity,
displacement and final structure moments are calculated from the
ICFM Equations. Control flow proceeds to step 606.
[0053] In step 606, the increment values calculated in step 604 are
summed to produce a total sum value of all calculated time
increment values. Control flow proceeds to step 608.
[0054] In step 608, frame stiffness values are adjusted for a next
increment calculation. Control flow proceeds to step 610 where the
results for each increment are scaled to the time of an event.
Processing continues as detailed above until ICFM calculations of
all time increments of the ground motion have been completed and
summed Steps 602-610 are further detailed in FIG. 6B
[0055] The ICFM equations given below are for an exemplary bridge
frame structure having C number of internal joints, and provide
final member-end-moments for the span just to the right (for right
equations) or left (for left equations) of an exemplary Joint A,
from a unit moment applied to exemplary Joint B. They also give
final column member-end-moments. In addition to the general right
and left equations, simplified equations are provided for the last
and first internal joints, but are not required as they give the
same results as the general equations. Thus, there are a total of
four independent ICFM equations if left and right expressions are
considerate separately. However, because the left and right
expressions are symmetric (left equations can be derived directly
from symmetry of the right equations), only two unique closed-form
equations may be implemented, one for the superstructure and the
other for the columns.
Right Moments
[0056] A Superstructure Right Moment is defined as
AB C = r B . r A ( - 2 ) A - B R B R C T C T A + 1 [ 1 - t A + 1 4
T A + 1 ] ( Equ . 1 ) ##EQU00001##
[0057] A Column Right Moment is defined as
AB C = r B . r A - 1 C A ( - 2 ) A - B R B R C T C T A + 1 ( Equ .
2 ) ##EQU00002##
[0058] A Simplified Superstructure Right Moment for a Last Internal
Joint is defined as
CB C = r B . r C ( - 2 ) C - B R B R C T C T A + 1 ( Equ . 3 )
##EQU00003##
and
[0059] A Simplified Column Right moment for a Last Internal Joint
is defined as
CB C = r B . r C - 1 c c ( - 2 ) C - B R B R C ( Equ . 4 )
##EQU00004##
Left Moments
[0060] A Superstructure Left Moment is defined as
BA C = t A . t B ( - 2 ) A - B T 1 T A R 1 R B - 1 [ 1 - r B - 1 4
R B - 1 ] ( Equ . 5 ) ##EQU00005##
[0061] A Column left Moment is defined as
BA C = t A . t B + 1 c B ( - 2 ) A - B T 1 T A R 1 R B - 1 ( Equ .
6 ) ##EQU00006##
[0062] A Simplified Superstructure Left Moment for a First Internal
Joint is defined as
1 A C = t 1 . t A ( - 2 ) A - 1 T 1 T A ( Equ . 7 )
##EQU00007##
And
[0063] A Simplified Column Left Moment For a First Internal Joint
is defined as
1 A C = t 2 . t A c 1 ( - 2 ) A - 1 T 1 T A ( Equ . 8 )
##EQU00008##
[0064] wherein: [0065] R=Cycle factor going to the right of the
beam [0066] T=Cycle factor going to the left of the beam [0067]
r=Distribution factor for member on the right side of a joint
[0068] t=Distribution factor for member on the left side of a joint
[0069] c=Distribution factor for column at a joint [0070]
AB.sub.C=Member moment just to the right of Joint A from a unit
moment applied at Joint B for a continuous beam or bridge frame
with C number of internal joints [0071] BA.sub.C=Member moment just
to the left of Joint B from a unit moment applied at Joint A for a
continuous beam or bridge frame with C number of internal joints
[0072] r.sub.Ar.sub.B=Multiplication of r.sub.A, r.sub.A+1, . . .
through r.sub.B [0073] r.sub.2r.sub.5=Multiplication of r.sub.2,
r.sub.3, r.sub.4 and r.sub.5 [0074] R.sub.AR.sub.B=Multiplication
of R.sub.A, R.sub.A+1, . . . through R.sub.B [0075]
t.sub.At.sub.B=Multiplication of t.sub.A, t.sub.A+1, . . . through
t.sub.B [0076] T.sub.AT.sub.B=Multiplication of T.sub.A, T.sub.A+1,
. . . through T.sub.B [0077] Note: if B equals A then
r.sub.Ar.sub.B=r.sub.A, t.sub.At.sub.B=t.sub.A,
T.sub.AT.sub.B=T.sub.A, R.sub.AR.sub.B=R.sub.A
[0078] FIG. 6B is an exemplary detailed flowchart illustrating a
method for the incremental closed-form Method (ICFM)
computationally efficient nonlinear structural analysis 600B. The
algorithm begins in step 632 where the closed-from equations are
used to determine no-sway and sway dead load forces. Processing
continues with step 634.
[0079] In step 634, the total column and superstructure moments are
found from dead load by summing the no-sway and sway results.
Processing continues with step 636.
[0080] In step 636, a lateral unit displacement is applied with
internal joints fixed from rotation, developing fixed-end-moments.
Then the closed-form equations are used to calculate final moments
for the unit sway. The column base shears are found from these end
moments and statics, and then summed to determine the total applied
force associated with the unit lateral displacement. The initial
frame stiffness is this force divided by the lateral displacement.
Processing continues with step 638.
[0081] In step 638, the ground acceleration is applied for a given
time increment. Processing continues with step 644.
[0082] In step 644, the stable average acceleration method is used
to determine the change in displacement, velocity and acceleration
of the frame over the time increment. Processing continues with
steps 646 and steps 648. In step 648, total displacement, velocity
and acceleration are summed before proceeding to step 650 to watch
for frame reversal (only if in a nonlinear state at time of
reversal). When the velocity goes through zero, the frame has
reversed and the analysis control then continues with step 654 as
all plastic hinges have a new rigid stiffness. The time of reversal
is directly calculated without iteration.
[0083] In step 646, the changes to column fixed-end-moments are
found based on the displacement increment from step 644. Processing
continues with step 640.
[0084] In step 640, the final incremental column and superstructure
moments are determined from application of the closed-form
equations to the incremental fixed-end-moments found in step 646.
Processing continues with step 642.
[0085] In step 642, total column and superstructure moments are
found by summing all prior incremental values up to that point in
time. Processing continues with step 652.
[0086] In step 652, the column moments from step 642 are compared
to the moment capacities at the column ends (plastic moments of the
columns). If all of the column moment demands are smaller than the
column moment capacities, then there is no nonlinear behavior and
the analysis continues by returning to step 638 in order to process
a next increment. Otherwise, if one or more of the column plastic
hinges develop (moment demand is greater than capacity in step
652), processing continues with step 654 defined below In step 654
the rotational stiffness of the plastic hinge (moment-rotation
spring) is set to zero if a plastic hinge has formed or made rigid
if a reversal has occurred. Processing continues with step 656.
[0087] In step 656, an additional full time increment step is taken
so that the algorithm can determine the new stiffness of the frame
with the moment-rotation stiffness of one (or more) hinges changed
in step 654. The new time increment to the time of the nonlinear
event (formation of a plastic hinge or reversal) is also determined
in step 656. Processing continues with step 658.
[0088] Step 658 scales all results back to the time of the
nonlinear event, including forces, frame displacement, frame
velocity and frame acceleration, as well as ground accelerations
and processing returns to step 638.
[0089] Processing continues as detailed above until ICFM
calculations of all time increments of ground motion have been
completed and summed.
[0090] FIGS. 7-9 provide an exemplary nonlinear time-history
analysis of a bridge frame using a computationally efficient
nonlinear structural analysis algorithm.
[0091] FIG. 7 illustrates a bridge frame in an exemplary nonlinear
time-history analysis using a computationally efficient nonlinear
structural analysis algorithm. A 5-span, pre-stressed concrete
bridge frame 700 has a total length of 720 ft. (219 m), with span
lengths of 120 ft, 150 ft and 180 ft and column lengths (702a-d) of
40 ft. and 50 ft.
[0092] FIG. 8 details an exemplary bridge frame superstructure
cross-section geometry in the exemplary nonlinear time-history
analysis. The box-girder superstructure is 6 feet 6 inches (1.98
meters) deep, 40 ft (12.2 meters) wide 800, having three cells
802-806. Overhangs 808-810 are 4 ft (1.22 m) wide and vary in
thickness from 8'' (0.203 m) at the edge-of-deck to 1 ft (0.305 m)
at the girder face. Reinforced concrete columns are circular with
5'-6'' (1.68 m) diameter.
[0093] FIG. 9 details exemplary column plastic moments for the
exemplary nonlinear time-history analysis of a bridge frame 700.
Plastic moments 902a-h vary from column-to-column 702a-d due to
different axial loads and primary steel percentages. The exemplary
concrete is normal-weight (unit weight of 150 pcf (23.6
kN/m.sup.3)), with design strength of 4 ksi (27.6 MPa), and
increased strength of 5 ksi (34.5 MPa) at the time of a future
earthquake. Using the American Concrete Institute (ACI) equation
for modulus of elasticity E, which is based on concrete strength,
this value is found to be E=4031 ksi (1,230 MPa). Cracked
properties are used for the columns, with cracked moment of inertia
taken to be 50% of the gross section moment of inertia. For the
pre-stressed bridge superstructure, 100% of the gross moment of
inertia is used. Table 1 provides moment of inertia and
cross-sectional area values for the columns and superstructure.
TABLE-US-00001 TABLE 1 Column and Superstructure Cross-Sectional
Properties Bridge I.sub.g (ft.sup.4)/ Area (ft.sup.2)/ I.sub.e
(ft.sup.4)/ Component (m.sup.4) (m.sup.2) (m.sup.4) Column
44.92/0.388 23.76/2.21 22.46/0.194 Superstructure 439.0/3.79
67.67/6.29 439.0/3.79
[0094] Linear-elastic and nonlinear exemplary analyses were
conducted using the proposed ICFM and the stiffness method to
compare analysis results, as well as run times. The stiffness
method is represented by the commercially available computer
program SAP2000. For a fair comparison between the two methods, all
input values, hysteresis models and assumptions are identical
between the model used in SAP2000 and the model developed for the
incremental closed-form approach. Axial and shear deformations are
constrained in SAP2000, resulting in the same number of
displacement degrees-of-freedom (seven displacement DOF; six
rotations and one translation) and mass degrees-of-freedom (single
translational mass DOF) as in the ICFM model. Also, the average
acceleration method is used in both solution schemes, with 2%
equivalent viscous damping. The fundamental natural period of the
cracked bridge frame is T.sub.n=1.23 s. The distributed
superstructure self-weight is found by multiplying the unit
concrete weight (150 pcf (23.6 kN/m.sup.3)) by its cross-sectional
area, giving
w.sub.s=(0.150)(67.67)=10.15 kips/ft (148 kN/m) (Equ. 9)
In addition to the bridge self-weight, the superstructure carries
two Type 25 barriers [13], one at the edge of each overhang, that
weigh 0.392 kips/ft (5.72 kN/m) each. Thus, the total distributed
weight along the bridge length is
w.sub.T=10.93 kips/ft (159 kN/m) (Equ. 10)
Bent caps are regions of the superstructure, above the columns,
that are solid and not cellular. For this bridge the width of the
bent cap is 6 ft (1.83 m), slightly larger than the column size.
The weight per bent cap, beyond the hollow superstructure weight
already accounted for, is W.sub.BC=132.3 kips (588 kN). Column
distributed weight is w.sub.c=3.564 kips/ft (52.0 kN/m), found from
the unit concrete weight and column cross-sectional area. For
inertial purposes in the dynamic analysis, it is assumed that half
of the column mass goes to ground and the other half is included
with the superstructure mass. Therefore, the total weight to be
considered when calculating the mass is
W.sub.B=(10.93)(720)+(132.3)(4)+(3.564)(40+50+50+40)/2=8720 kips
(38,800 kN) (Equ. 11)
Allowing the mass of the bridge to be determined as
M.sub.B=8720/32.2=270.8 kip*s.sup.2/ft (3,950 kg) (Equ. 12)
[0095] FIGS. 10-20 detail the results of subjecting the exemplary
bridge frame superstructure 700 provided in FIGS. 7-9 to a recorded
earthquake motion from the 1989 Loma Prieta earthquake using the
computationally efficient nonlinear structural analysis algorithm.
FIGS. 10-12 show the measured ground behavior, with the measured
acceleration time-history (FIG. 10) used as input for the analysis.
It was measured at the Capitola Fire Station in the East/West
direction. Acceleration and displacement profiles are given in
FIGS. 10 (1000) and 11 (1000), respectively. Peak ground
acceleration is just below 0.5 g. A 5%-damped, linear-elastic
acceleration response spectrum is provided in FIG. 12 (1200). The
measured ground motion has duration of 40 s with time increments of
0.02 s, resulting in 2,000 increments.
[0096] FIGS. 13-20 detail results from the analysis of the present
computationally efficient nonlinear structural analysis algorithm
applied to the exemplary structure.
[0097] FIGS. 13 and 14 show the linear-elastic (FIG. 13) and
nonlinear (FIG. 14) responses of the bridge frame, with total force
on the y axis and relative frame displacement on the x axis.
Relative displacement is the difference between the absolute
displacements of the ground and the frame. The force is the sum of
the column shears transferred to the ground (base shear). All of
the figures show that the two methods (ICFM and stiffness method)
provide the same results. FIGS. 15 and 16 give linear-elastic and
nonlinear relative displacement results over time, with the x axis
being time and the y axis relative displacement. FIGS. 17 and 18
provide linear-elastic and nonlinear results for frame base shear
(y axis) versus time (x axis). Nonlinear results are also shown in
FIG. 17 to demonstrate how much larger the linear-elastic force
results are. In FIG. 18 the total force capacity of the frame is
given as dotted lines (at plus and minus 1500 kips), with all
plastic hinges developed. This shows that the nonlinear analysis
scheme does not exceed the strength of the frame. Zoom-in results
(between 5 and 15 s) for nonlinear displacements and forces are
given in FIGS. 19 and 20, again demonstrating that the results from
the new ICFM are the same as from the stiffness method.
[0098] The total nonlinear run-time for the exemplary bridge frame
and ground motion using the stiffness method was 204.8 s compared
to 0.0624 s from the present incremental closed-form approach. A
ratio of these analysis times shows that for the exemplary case,
the computationally efficient nonlinear structural analysis
algorithm is more than 3,000 times faster than the stiffness method
represented in SAP2000 as shown in Table 2.
TABLE-US-00002 TABLE 2 Nonlinear Time-History Solution Times for
ICFM and Stiffness Method Stiffness Closed- Time Method Form Ratio
Duration Increment Number of Solution Solution of Earthquake File
(s) (s) Increments Time Time Times 1989 Loma 40 0.02 2,000 204.8 s
0.06240 s 3,282 Prieta EQ, (3.41 min) Capitola Fire Station E/W Ch3
2010 Mexicali 50 0.005 10,000 1,410 s 0.2808 s 5,021 EQ, Calexico -
(23.5 min) El Centro Array 11 N/S Ch1
[0099] Because there were 2,000 increments in the ground motion,
this dramatic increase in speed indicates that the ICFM would be
finished with the full earthquake duration before a single time
increment is assessed by the traditional stiffness method. Results
are given in FIGS. 13-20 for both the present ICFM and the
traditional stiffness method, as discussed previously. Because the
values are almost identical, the graphs are nearly
indistinguishable from one another. Linear-elastic results are
given in separate FIGURES from the nonlinear results of interest.
Hysteretic force-deformation behaviors of the bridge frame for
linear-elastic and nonlinear responses are given in FIGS. 13 (1300)
and 14 (1400), respectively.
[0100] As expected, the linear-elastic response stays on the same
line and always goes through the origin, with no residual
displacement (1300). However, once plastic hinges have formed, the
nonlinear behavior of the bridge frame cycles around and results in
permanent plastic displacements (1400).
[0101] FIG. 14 details the exemplary bridge frame computationally
efficient structural analysis algorithm nonlinear
force-displacement bridge frame results from the 1989 Loma Prieta
earthquake. The strength of the exemplary bridge frame is 1,500
kips (6,670 kN) with all plastic hinges developed, achieved in both
loading directions (1400). FIG. 14 also shows that nonlinear
behavior occurs in each direction at forces that are much lower
than the full strength of the frame. Relative displacement
time-history results are provided in FIGS. 15 and 16 for
linear-elastic (1500) and nonlinear responses (1600), respectively.
Once plastic hinges have formed, the nonlinear behavior is quite
different from the linear behavior, having permanent offset and
different maximum values.
[0102] Base shear results over time are given in FIGS. 17 and 18
for linear (1700) and nonlinear (1800) cases. From FIG. 18 it is
clear that the strength of the frame is reached in both directions,
while from linear-elastic analysis, the maximum force demand is
8,315 kips (37,000 kN), which is 5.54 times higher than from
nonlinear analysis and 5.54 times larger than the force capacity of
the frame (1,500 kips (6,670 kN)).
[0103] The nonlinear force response is given on the same graph as
the linear-elastic force response (1700) in FIG. 17, along with the
frame force capacity lines drawn at plus and minus 1,500 kips
(6,670 kN). This succinctly illustrates that the large forces from
a linear-elastic analysis are not physically possible due to
strength limits of the frame associated with column plastic
hinging. Maximum displacements from linear-elastic (1500) and
nonlinear (1600) analyses are 1.19 ft (0.363 m) and 0.702 ft (0.214
m), respectively. This shows that for the exemplary bridge frame
structure, and chosen 1989 Loma Prieta earthquake motion, the
displacement demand from NTHA is only 59% of the linear-elastic
demand, indicating possible cost savings from the more detailed
nonlinear analysis.
[0104] Zoomed-in views (from 5 to 15 s) of nonlinear displacement
(1900) and force (2000) results show that the ICFM provide the same
maximum and time-history values as the stiffness method illustrated
by FIGS. 19 and 20, respectively. Advantageously however, as
indicated in Table 2, the present ICFM performs at orders of
magnitude faster speed than the stiffness method. For the 1989 Loma
Prieta earthquake motion discussed in the example, the ICFM was
more than 3,000 times faster than the stiffness method. When a
different motion from the 2010 Mexicali earthquake was used (Table
2), with five times the number of increments than the original
motion, the ICFM was more than 5,000 times faster than the
stiffness method. In this case the stiffness method took over 23
minutes while the ICFM took just over a quarter of a second.
[0105] In conclusion, a completely novel approach has been
developed to assess the nonlinear time-history response of a bridge
frame or other structure subjected to seismic loading. The present
ICFM has proven to be thousands of times faster than the
conventional stiffness method while providing the same maximum
results and the same results over time. Identical input models were
created based on the example bridge provided and the discussions
given in the text. All nonlinear time-history results were the same
between the two methods, with the only difference being the time to
compute these results; the ICFM was over 3,000 times faster than
the stiffness method for the primary ground motion studied and over
5,000 times faster for a second ground motion, which had more
increments due to a smaller time step and longer duration.
[0106] Such increases in computational speed will result in changes
to the way bridge frame structures are designed for seismic
loading. Multiple earthquake motions can now be calculated through
the bridge frame or other structure, allowing all possible
scenarios to be considered by the bridge designer. Using the new
ICFM, the average NTHA solution time for the two different
earthquake motions was 0.172 s for the example bridge. Taking this
as an approximate average for multiple motions, the nonlinear
bridge response from 10 different earthquake motions can be
assessed in less than two seconds, with maximum results of interest
automatically saved for later viewing. Following the same logic,
over 300 different nonlinear time-history analyses could be
performed in one minute for the 5-span bridge frame example. Using
traditional stiffness method (represented by the computer program
SAP2000) this same task would take over 67 hours of straight
running time.
[0107] While, theoretically, the stiffness method or the ICFM could
be used to perform NTHA of a bridge frame, due to practical
limitations the stiffness method has not been used for everyday
seismic bridge design, relegating it to large, high budget bridge
projects. The speed, stability and ease of use of the new ICFM
allows NTHA to be conducted for seismic design of commonplace
bridge frame or other structures. Furthermore, the algorithm has
been specifically designed for this purpose with minimal input
required. Because it will provide a proper assessment tool to
bridge design engineers for seismic demands, the ICFM makes bridge
structures safer and, in many cases, significantly reduces bridge
construction costs.
[0108] For the example bridge frame and chosen earthquake motion,
NTHA resulted in 59% of the displacement demand from linear-elastic
analysis, indicating over-design and increased costs if the larger
displacement demands from linear-elastic analysis are used for
seismic design. In other cases it can be the opposite, with NTHA
displacement demands considerably larger than from linear-elastic
analysis. In this case the bridge will be seismically unsafe if the
smaller displacement demands from linear-elastic analysis are
targeted in design. While there are various linear-elastic methods
that attempt to capture equivalent nonlinear seismic displacement
demands of a bridge frame, they are not capable of providing
consistent results that only NTHA can give.
[0109] Other embodiments of the ICFM algorithm comprise 3-D frame
effects, different hysteresis models for column plastic hinges,
P-delta effects, soil springs at the base of the columns,
interaction between adjacent bridge frames as well as between a
bridge frame and the soil behind abutments.
[0110] Those of skill in the art would understand that information
and signals may be represented using any of a variety of different
technologies and techniques. For example, data, instructions,
commands, information, signals, bits, symbols, and chips that may
be referenced throughout the above description may be represented
by voltages, currents, electromagnetic waves, magnetic fields or
particles, optical fields or particles, or any combination
thereof.
[0111] Those of skill would further appreciate that the various
illustrative logical blocks, modules, circuits, and algorithm steps
described in connection with the embodiments disclosed herein may
be implemented as electronic hardware, computer software, or
combinations of both. To clearly illustrate this interchangeability
of hardware and software, various illustrative components, blocks,
modules, circuits, and steps have been described above generally in
terms of their functionality. Whether such functionality is
implemented as hardware or software depends upon the particular
application and design constraints imposed on the overall system.
Skilled artisans may implement the described functionality in
varying ways for each particular application, but such
implementation decisions should not be interpreted as causing a
departure from the scope of the present invention.
[0112] The various illustrative logical blocks, modules, and
circuits described in connection with the embodiments disclosed
herein may be implemented or performed with a general purpose
processor, a digital signal processor (DSP), an application
specific integrated circuit (ASIC), a field computationally
efficient non-linear structural analysis algorithm mable gate array
(FPGA) or other computationally efficient non-linear structural
analysis algorithm mable logic device, discrete gate or transistor
logic, discrete hardware components, or any combination thereof
designed to perform the functions described herein. A general
purpose processor may be a microprocessor, but in the alternative,
the processor may be any conventional processor, controller,
microcontroller, or state machine. A processor may also be
implemented as a combination of computing devices, e.g., a
combination of a DSP and a microprocessor, a plurality of
microprocessors, one or more microprocessors in conjunction with a
DSP core, or any other such configuration.
[0113] The steps of a method or algorithm described in connection
with the embodiments disclosed herein may be embodied directly in
hardware, in a software module executed by a processor, or in a
combination of the two. A software module may reside in RAM memory,
flash memory, ROM memory, EPROM memory, EEPROM memory, registers,
hard disk, a removable disk, a CD-ROM, or any other form of storage
medium known in the art. An exemplary storage medium is coupled to
the processor such that the processor can read information from,
and write information to, the storage medium. In the alternative,
the storage medium may be integral to the processor. The processor
and the storage medium may reside in an ASIC. The ASIC may reside
in a user terminal. In the alternative, the processor and the
storage medium may reside as discrete components in a user
terminal.
[0114] In one or more exemplary embodiments, the functions
described may be implemented in hardware, software, firmware, or
any combination thereof. If implemented in software, the functions
may be stored on or transmitted over as one or more instructions or
code on a computer-readable medium. Computer-readable media
includes both computer storage media and communication media
including any medium that facilitates transfer of a computer
computationally efficient nonlinear structural analysis algorithm
from one place to another. A storage media may be any available
media that can be accessed by a computer. By way of example, and
not limitation, such computer-readable media can comprise RAM, ROM,
EEPROM, CD-ROM or other optical disk storage, magnetic disk storage
or other magnetic storage devices, or any other medium that can be
used to carry or store desired computationally efficient nonlinear
structural analysis algorithm code in the form of instructions or
data structures and that can be accessed by a computer. Also, any
connection is properly termed a computer-readable medium. For
example, if the software is transmitted from a website, server, or
other remote source using a coaxial cable, fiber optic cable,
twisted pair, digital subscriber line (DSL), or wireless
technologies such as infrared, radio, and microwave, then the
coaxial cable, fiber optic cable, twisted pair, DSL, or wireless
technologies such as infrared, radio, and microwave are included in
the definition of medium. Disk and disc, as used herein, includes
compact disc (CD), laser disc, optical disc, digital versatile disc
(DVD), floppy disk and blu-ray disc where disks usually reproduce
data magnetically, while discs reproduce data optically with
lasers. Combinations of the above should also be included within
the scope of computer-readable media.
[0115] The previous description of the disclosed embodiments is
provided to enable any person skilled in the art to make or use the
present invention. Various modifications to these embodiments will
be readily apparent to those skilled in the art, and the generic
principles defined herein may be applied to other embodiments
without departing from the spirit or scope of the invention. Thus,
the present invention is not intended to be limited to the
embodiments shown herein but is to be accorded the widest scope
consistent with the principles and novel features disclosed
herein.
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